The proof of the Goldbach's Conjecture is one of the biggest still unsolved problems regarding prime numbers. Originally expressed in 1742 by the mathematician Christian Goldbach, from whom the conjecture takes its name, it was rephrased by Euler in the form in which we know it today:

**Every even number greater than 2 can be expressed as a sum of two prime numbers.**

In spite of the empirical evidence and the simplicity of its statement, the conjecture is resisting to every attempts of proof since almost three centuries. Several mathematicians have proved some weaker versions of the conjecture.

The statement phrased by Christian Goldbach is a "conjecture", so, as a matter of principle, it is a hypothesis. This means that it can be:

- True, that is all even numbers greater than 2 can be expressed as a sum of two prime numbers;
- False, that is at least one even number greater than two exists, which cannot be written as a sum of two prime numbers.

Currently, there are several attempts to prove the Goldbach's conjecture, which are complete, in the sense that they come to the conclusion, but contain a series of problems, so they have not been considered valid by the international community. We instead propose some proof strategies, which are still far from being complete, but already contain interesting and unrefuted (so far) intermediate results. We hope this material will be a useful starting point for who, like us, has started looking for a proof...

- True, that is all even numbers greater than 2 can be expressed as a sum of two prime numbers;
- False, that is at least one even number greater than two exists, which cannot be written as a sum of two prime numbers.

Goldbach's conjecture is put into the field of Number theory, the branch of Mathematics which studies the properties of integer numbers. In order to understand the proofs of the results similar to the conjecture, and very likely also to prove the conjecture itself, a sound knowledge of number theory is required. But this kind of knowledge rarely is part of a mathematician's curriculum...

Dashed line theory is a new mathematical theory which studies the connection between the sequence of natural numbers and their divisibility relationship. Typical problems are the computation of the n-th natural number divisible by at least one of k fixed numbers, or not divisible by any of them. Due to this nature, the theory is suited for studying prime numbers with a constructive approach, inspired to the sieve of Eratosthenes...

# Latest posts

Looking at a prime numbers table, it's very simple to notice how their distribution seems to escape any regularity; instead it's more difficult to note how behind this ostensible irregularity a precise order hides itself. This order is given by the prime number Theorem. The proof of this theorem is interesting in itself, also from an historical point of view.

In this post we'll revisit Chebyshev's Theorem, according to which the function π(x), that counts the number of prime numbers less than or equal to a positive integer x, has the same order of magnitude as x/(log x). We'll see that not only the two functions have the same order of magnitude, but also that, if a limit for their ratio exists, that limit must be 1.

In this post we'll see two concepts of mathematical analysis which will be useful in number theory: the limit inferior (lim inf) and the limit superior (lim sup) of a sequence. We'll see that they can be defined starting from the study of the intervals that contain all, or almost all, the terms of the sequence.

Almost certainly you already know the factorial function, indicated by x!, which is read as "x factorial" and for an integer x > 0 it consists in the product of all positive integers up to x. In this post we'll study this function, that often occurs in number theory, from two different points of view: first we'll study its order of magnitude; then we'll decompose the product x! in such a way to highlight the prime factors which it's made up of.

In this post we'll see a technique that will let us overestimate or underestimate a value assumed by a function defined on integer numbers, by means of an appropriate integral of a monotonous extension of it. We'll apply this technique to a logarithmic function, obtaining a lemma that will turn out to be useful in the future.